XV.] DEDUCTIVE REASONING. S.-^ 



For the term "matter" in eitlier of these identities we 

 may substitute its equivalent given in the ot]i(>r definition. 

 Elsewhere they often employ sentences of the form exeui- 

 plified in the following:^ "The integral curvature, or 

 whole change of direction of an arc of a plane curve, is the 

 angle through which the tangent has turned as we pass from 

 one extremity to the other." This sentence is certainly of 

 the form — 



The integral curvature — the whole change of direc- 

 tion, &c. = the angle through which the tangent 

 has turned, &c. 



Disguised cases of the same kind of inference occur 

 throughout all sciences, and a remarkable instance is found 

 in algebraic geometry. j\Iathematicians readily show that 

 every equation of the form y = mx -f c corresponds to or 

 represents a sti'uight line ; it is also easily proved that the 

 same equation is equivalent to one of the general ibrni 

 Kx + By + C = o, and vice, versd. ITen(^e it follows that 

 every equation of the form in question, that is to say, 

 every equation of the first degree, corresponds to or 

 represents a straight line."'' 



Inference loith a Simple and a Pcirtial Ide-ntHy. 



A form of reasoning somewhat different fronj tiiat hist 

 considered consists in inference between a simph; nud a 

 partial identity. If we have two propositions of the lorms 



A = 1^., 

 B = BC, 

 we may then substitute for B in either proposition its 

 equivalent in the other, getting in both cases A =-- P>C ; 

 in this we may if we like make a second substitution for 

 ]}, getting 



A = AC. 

 ^ Thus, since " Tlie IVfont Blanc is the highest mountniii in 

 Einope, and the Mont Blanc is deeply covered with i^novv,'"' 

 we infei' liy an obvicnis substitution that " Tlie highest 

 mountain in Euro))e is deeply covered with snow." These 

 propositions when rigorously stated fall into the forms 

 above exhibited. 



This mode of inference is constantly ei-iployed wlien for 



' Treatue on Natnral fJiilonophy, vol. i. p. 6. 



* 1<jdh:uit(.T's riaue Co-wdina'e GcoviHiy, chuj, ii. [.p. xi —14. 



